To slaughter can be the process of killing animals for meat, as in a "slaughterhouse", or "led to the slaughter". It also is used to describe any large scale and often senseless killing of humans or animals, as in a particularly bloody battle.
After the slaughter at Wounded Knee, soldiers buried at least 200 deceased Lakota in a mass grave.
To butcher is somewhat synonymous with "to slaughter"; however while slaughter refers mostly to the killing, butcher refers to the cutting up of animal carcasses for sale and consumption.
Later in the year, the animals were efficiently butchered, and various parts hung in the smokehouse to cure.
As with "slaughter", "butcher" can refer to killing humans, often either with particular cruelty or where one side does most of the killing.
The guards butchered many hundreds of prisoners before the invading army could arrive to free them.
To cut is a generic term which means to use a blade of some kind to make an incision, or to divide something into parts ("to cut up"). It does not specifically refer to processing animals for meat, although it would be possible to say something like:
After cleaning the deer, the hunters cut up the meat into smaller portions so it would be easier to carry.
You can cut/cut up almost anything: fruit, clothing, wood, and of course humans:
The soldiers launched a surprise attack in the middle of the night and cut their enemy to shreds.
This use of "cut" is somewhat less violent than "slaughter" or "butcher".